Only six third round ties could be completed on Sunday after rain caused an early cancellation.
Asked if he rated it his best performance of the season, Murray said: "I played some good matches at the beginning of the year". He progressed to win the match 7-6 (8) 7-5 6-0, snagging the final eight games having briefly faltered in dropping serve when serving for the second set at 5-4.
Verdasco, world no. 37, needed one hour and 33 minutes to defeat his Uruguayan rival 6-2, 6-1, 6-3. "You never know when you can win a set or a match, and that's happen today". "I'm working things out while I'm playing the matches, and that's a really, really important part of my game". I think he's done a really great job.
But the Argentinian then double faulted before getting a fourth set point, on which he missed a forehand.
The pair traded blows in the second set with Murray creeping ahead but a monumental effort by del Potro saw him break back for 5-5, only to instantly drop serve again as Murray yanked him around the court with tight angles, drop shots and lobs.
The 28-year-old slumped over the net and stayed bent over during most of the changeover to catch his breath.
Del Potro, who beat Murray the last time they met at the Davis Cup in September past year, was struggling with a groin injury sustained in his last match but broke serve just three games in to take a 3-1 lead. With some exquisite returning and acutely angled groundstrokes, Murray raced through the final set and into the fourth round.
He knew the opener was key in the slow, heavy conditions and by the third set, he was on the ropes, winning just 11 points.
Touching performances at Ariana Grande's Manchester concert
Mr Burnham appeared on stage with representatives of political parties, religions and both of the city's football clubs. Fellow fans Jasmine Mayers and Lydia Smith said they came to the concert "to show support to the victims' families".
Murray started in the same vein, breaking Del Potro at the start of the third an this time ploughing on as he denied Del Potro a chance of a game at all in the third.
Stan Wawrinka, the 2015 champion, was a 7-6 (2), 6-0, 6-2 victor over 28th-seeded Fabio Fognini.
Croatian seventh seed Marin Cilic also advanced after sweeping aside Spain's Feliciano Lopez 6-1, 6-3, 6-3 to reach the last 16 without dropping a set.
The left-hander blasted 31 winners as he took just 93 minutes to dismiss Uruguayan 22nd seed Pablo Cuevas 6-2 6-2 6-3 and book a likely showdown with Japanese star Kei Nishikori.
In the women's event, France have three women in the last 16 for the first time in 23 years. Cornet and Caroline Garcia, due to meet in the next round, are barely on speaking terms after a bitter feud.
Carina Witthoeft turned up late for her third round match against second-seeded Karolina Pliskova at the French Open. Garcia fell out with her teammates after severing her successful doubles relationship with Kiki Mladenovic, before pulling out of Fed Cup duty in April.
"I'm starting to feel better", Murray said. I think we have a great team.
She will now face Spain's Carla Suarez Navarro for a place in the quarter-finals after the number 21 seed claimed a 6-4, 6-4 win over Elena Vesnina of Russian Federation.
Former top-ranked Caroline Wozniacki made it into the fourth round after beating American teenager CiCi Bellis when their match resumed.